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06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 9 Home Quit Lesson 6.2 Exercises, pages 448–458 A 3. Sketch each angle in standard position. a) 75° b) 140° Since the angle is between 0° and 90°, the terminal arm is in Quadrant 1. Since the angle is between 90° and 180°, the terminal arm is in Quadrant 2. y y 75° 140° x c) 200° d) 340° Since the angle is between 180° and 270°, the terminal arm is in Quadrant 3. Since the angle is between 270° and 360°, the terminal arm is in Quadrant 4. y y 200° x x O ©P x O O DO NOT COPY. 340° O 6.2 Angles in Standard Position in All Quadrants—Solutions 9 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 10 Home Quit 4. a) Determine the reference angle for each angle in standard position. i) 34° ii) 98° The angle is in Quadrant 1, so its reference angle is 34°. iii) 241° The angle is in Quadrant 2, so its reference angle is: 180° 98° 82° iv) 290° The angle is in Quadrant 3, so its reference angle is: 241° 180° 61° The angle is in Quadrant 4, so its reference angle is: 360° 290° 70° b) For each angle in part a, determine the other angles between 90° and 360° that have the same reference angle. i) 34° ii) 98° The angles with the same reference angle are: 180° 34° 146° 180° 34° 214° 360° 34° 326° iii) 241° The angles with the same reference angle are: 180° 82° 262° 360° 82° 278° iv) 290° The angles with the same reference angle are: 180° 61° 119° 360° 61° 299° The angles with the same reference angle are: 180° 70° 110° 180° 70° 250° 5. Determine the quadrant in which the terminal arm of each angle in standard position lies. a) 280° Since the angle is between 270° and 360°, the terminal arm lies in Quadrant 4. c) 191° Since the angle is between 180° and 270°, the terminal arm lies in Quadrant 3. 10 b) 88° Since the angle is between 0° and 90°, the terminal arm lies in Quadrant 1. d) 103° Since the angle is between 90° and 180°, the terminal arm lies in Quadrant 2. 6.2 Angles in Standard Position in All Quadrants—Solutions DO NOT COPY. ©P 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 11 Home Quit B 6. Point P(⫺3, 4) is a terminal point of an angle u in standard position. P(⫺3, 4) a) Sketch the angle. Plot P(3, 4); draw a line through OP. Label U. Let the length of OP r. y r x O b) What is the distance from the origin to P? √2 2 Use: r x y Substitute: x 3, y 4 r √ (3)2 42 √ r 25 r5 c) Write the primary trigonometric ratios of u. x 3, y 4, r 5 y x sin U r cos U r 3 3 , or 4 5 5 y tan U x 5 4 4 , or 3 3 d) To the nearest degree, what is u? Use: sin U 4 5 The reference angle is: sin1 a b 53.1301. . .° 4 5 U is approximately 180° 53°, or 127°. 7. Each angle u is in standard position. State the quadrants in which the terminal arm of the angle could lie. 2 a) cos u = - 3 x cos U r , so x is negative; and the terminal arm lies in Quadrant 2 or 3. 1 c) sin u = 2 y sin U r , so y is positive; and the terminal arm lies in Quadrant 1 or 2. ©P DO NOT COPY. b) tan u = -4 y tan U x , so if x is negative, the terminal arm lies in Quadrant 2; if y is negative, the terminal arm lies in Quadrant 4. d) tan u = 0.25 y tan U x , so if both x and y are positive, the terminal arm lies in Quadrant 1; if both x and y are negative, the terminal arm lies in Quadrant 3. 6.2 Angles in Standard Position in All Quadrants—Solutions 11 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 12 Home Quit 8. a) Choose an angle between 90° and 180°. Sketch a diagram to show that the angle can be formed by reflecting its reference angle in an axis or axes. Sample response: For 115°, its reference angle is: 180° 115° 65° Draw 65° in Quadrant 1. Choose a terminal point, P. Reflect OP in the y-axis. Then OP’ is the terminal arm of 115°. P⬘ P y 115⬚ x O 65⬚ b) Repeat part a for an angle between 180° and 270°. Sample response: For 220°, its reference angle is: 220° 180° 40° P⬘ Draw 40° in Quadrant 1. Choose a terminal point, P. Reflect OP in the y-axis, then reflect OP’ in the x-axis. 220⬚ Then OP’’ is the P⬙ terminal arm of 220°. y P x 40⬚ O c) Repeat part a for an angle between 270° and 360°. Sample response: For 310°, its reference angle is: 360° 310° 50° Draw 50° in Quadrant 1. Choose a terminal point, P. Reflect OP in the x-axis. 310⬚ Then OP’ is the terminal arm of 310°. y P 50⬚ x O P⬘ 9. Determine possible coordinates of a terminal point for each angle in standard position. a) 315° Sample response: Choose a value for r, such as r 4. Then: x r cos 315° y r sin 315° 4 a√ b , or √ 1 2 4 2 4 a√ b , or √ 1 2 4 2 Possible coordinates are: a√ , √ b 4 2 12 4 2 6.2 Angles in Standard Position in All Quadrants—Solutions DO NOT COPY. ©P 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 13 Home Quit b) 210° Sample response: Choose a value for r, such as r 5. Then: x r cos 210° y r sin 210° √ √ 3 5 3 5 a b , or 2 2 5 a b , or 1 2 √ 5 3 5 , b Possible coordinates are: a 2 2 5 2 c) 120° Sample response: Choose a value for r, such as r 6. Then: x r cos 120° y r sin 120° √ √ 3 6 a b , or 3 3 2 √ Possible coordinates are: (3, 3 3) 1 6 a b , or 3 2 10. Each point below lies on the terminal arm of an angle u in standard position. Determine: i) cos u ii) sin u iii) tan u iv) the value of u to the nearest degree a) A(⫺3, ⫺4) √ r x2 y2 √ 2 r b) B(⫺6, 0) √ r x2 y2 √ 2 (3) (4)2 (6) (0)2 r r5 r6 x i) cos U r 3 y ii) sin U r 5 y iii) tan U x 4 4 , or 3 3 4 iv) Use: tan U 3 The reference angle is: tan1 a b 53.1301. . .° 4 3 x 4 5 y i) cos U r 6 ii) sin U r 1 0 6 0 6 y iii) tan U x 0 , or 0 6 iv) Since the terminal point lies on the negative x-axis, U 180° Since both x and y are negative, the terminal arm lies in Quadrant 3, and U is approximately: 180° 53° 233° ©P DO NOT COPY. 6.2 Angles in Standard Position in All Quadrants—Solutions 13 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 14 Home d) D(2, ⫺1) √ r x2 y2 √ r (2)2 (1)2 √ c) C(0, 2) √ r x2 y2 √ 2 r (0) (2)2 r2 r x i) cos U r 0 2 y ii) sin U r 2 2 0 1 y iii) tan U x Quit 2 0 tan U is undefined. iv) Since the terminal point is on the positive y-axis, U 90° 5 y x i) cos U r ii) sin U r 1 √ 2 √ 5 5 y iii) tan U x 1 1 , or 2 2 2 iv) Use: cos U √ 5 The reference angle is: cos1 a √ b 26.5650. . .° 2 5 Since x is positive and y is negative, the terminal arm lies in Quadrant 4 and U is approximately: 360° 27° 333° 11. A hiker sketches a map of her destination, D, from her starting point, O. The hiker can travel only west, then north. To the nearest kilometre, how far must she hike to get to her destination? N D 10 km 140⬚ W O E S The distance the hiker travels west is the absolute value of the x-coordinate of D. x r cos U Substitute: r 10 and U 140° x 10 cos 140° x 7.6604. . . The distance the hiker travels north is the y-coordinate of D. y r sin U Substitute: r 10 and U 140° y 10 sin 140° y 6.4278. . . In kilometres, the hiker travels: 7.6604. . . 6.4278. . . 14.0883. . . The hiker travels approximately 14 km. 14 6.2 Angles in Standard Position in All Quadrants—Solutions DO NOT COPY. ©P 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 15 Home Quit 12. Explain why (sin u)2 + (cos u)2 = 1 for any angle u in standard position in Quadrant 2. y x In Quadrant 2, cos U r and sin U r 2 y So, (sin U)2 (cos U)2 ar b a rb y2 x 2 x2 r r2 2 y x2 2 r2 Since y x r , 2 2 2 (sin U)2 (cos U)2 r2 r2 1 13. The location of the terminal arm of an angle a and a trigonometric ratio of its reference angle u are given. 1 a) The terminal arm lies in Quadrant 3, cos u = 4 ; what is tan a? Let Q(x, y) be a point on the terminal arm of U that is r units from O. cos U 1 x and cos U r , so write x 1 and r 4 4 Determine the value of y. r 2 x2 y2 42 12 y2 y √ 15 √ Then P(1, 15) lies on√the terminal arm of A. y tan A x , so tan A √ 15 , or 15 1 3 b) The terminal arm lies in Quadrant 2, tan u = 7 ; what is sin a? Consider point Q from part a. tan U y 3 and tan U x , so y 3 and x 7 7 Determine the value of r. √2 x y2 √ r 72 32 √ r r 58 Then P(7, 3) lies on the terminal arm of A. y 3 sin A r , so sin A √ 58 ©P DO NOT COPY. 6.2 Angles in Standard Position in All Quadrants—Solutions 15 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 16 Home Quit 5 c) The terminal arm lies in Quadrant 4, sin u = 12 ; what is cos a? Consider point Q from part a. sin U y 5 and sin U r , so y 5 and r 12 12 Determine the value of x. r2 x2 y2 122 x2 52 x √ 119 √ Then P( 119 , 5) lies on the terminal arm of A. √ x cos A r , so cos A 119 12 14. Which values of u satisfy each equation for 0° ≤ u ≤ 360°? a) tan u = 1 Use the table on page 444. U 45° and U 225° c) sin u = 1 Use the table on page 444. U 90° e) cos u = 1 Use the table on page 444. U 0° and U 360° b) tan u = 0 U 0°, U 180°, and U 360° d) sin u = 0 U 0°, U 180°, and U 360° f) cos u = 0 U 90° and U 270° 15. Which values of u satisfy each equation for 0° ≤ u ≤ 360°? a) tan u = ⫺1 Use the table on page 444. U 135° and U 315° b) sin u = ⫺1 U 270° c) cos u = ⫺1 Use the table on page 444. U 180° 16. To the nearest degree, which values of u satisfy each equation for 0° ≤ u ≤ 360°? 1 a) tan u = 2 The reference angle is: The reference angle is: tan1 a b ⬟ 27° tan1 a b ⬟ 34° tan U is positive in Quadrant 1, so U ⬟ 27° or in Quadrant 3, so U ⬟ 180° 27°, or 207° tan U is negative in Quadrant 2, so U ⬟ 180°34°, or 146° or in Quadrant 4, so U ⬟ 360° 34°, or 326° 1 2 16 2 b) tan u = ⫺3 2 3 6.2 Angles in Standard Position in All Quadrants—Solutions DO NOT COPY. ©P 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:11 PM Page 17 Home Quit d) sin u = ⫺0.25 c) cos u = 0.6 The reference angle is: cos1(0.6) ⬟ 53° cos U is positive in Quadrant 1, so U ⬟ 53° or in Quadrant 4, so U ⬟ 360° 53°, or 307° The reference angle is: sin1(0.25) ⬟ 14° sin U is negative in Quadrant 3, so U ⬟ 180° 14°, or 194° or in Quadrant 4, so U ⬟ 360° 14°, or 346° 17. a) Determine sin 40°. For which values of u, where 0° ≤ u ≤ 360°, is cos u = sin 40°? sin 40° 0.6427. . . Solve: cos U 0.6427. . . U cos1(0.6427. . .) U 50° cos U is also positive in Quadrant 4. U 360° 50°, or 310° b) Repeat part a for the sines of 4 different angles between 0° and 90°. What patterns do you see in the angles? Sample response: sin 22° 0.3746. . . Solve: cos U 0.3746. . . U 68° Also, U 360° 68°, or 292° sin 59° 0.8571. . . Solve: cos U 0.8571. . . U 31° Also, U 360° 31°, or 329° sin 76° 0.9702. . . Solve: cos U 0.9702. . . U 14° Also, U 360° 14°, or 346° sin 83° 0.9925. . . Solve: cos U 0.9925. . . U 7° Also, U 360° 7°, or 353° When the sine and the cosine of an angle are equal, the angles in Quadrant 1 are complementary. 18. a) Determine tan 40°. For which values of u, where 0° ≤ u ≤ 360°, 1 is tan u = ? tan 40° tan 40° 0.8390. . . Solve: tan U 1 , or 1.1917. . . 0.8390. . . U tan1(1.1917. . .) U 50° tan U is also positive in Quadrant 3. U 180° 50°, or 230° ©P DO NOT COPY. 6.2 Angles in Standard Position in All Quadrants—Solutions 17 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:12 PM Page 18 Home Quit b) Repeat part a for the tangents of 4 different angles between 0° and 90°. What patterns do you see in the angles? Sample response: tan 22° 0.4040. . . Solve: tan U tan 59° 1.6642. . . Solve: 1 , or 2.4750. . . 0.4040. . . tan U 1 , or 0.6008. . . 1.6642. . . U 68° Also, U 180° 68°, or 248° U 31° Also, U 180° 31°, or 211° tan 76° 4.0107. . . Solve: tan 83° 8.1443. . . Solve: tan U 1 , or 0.2493. . . 4.0107. . . tan U 1 , or 0.1227. . . 8.1443. . . U 14° U 7° Also, U 180° 14°, or 194° Also, U 180° 7°, or 187° The tangents of complementary angles are reciprocals. 19. For each equation below: i) To the nearest degree, determine the possible values of u, for 0° ≤ u ≤ 360°. ii) Determine the values of the other two primary trigonometric ratios of each angle u, to 3 decimal places. a) cos u = 0.2 i) cos1(0.2) ⬟ 78° cos U is positive in Quadrant 1, so U ⬟ 78° or in Quadrant 4, so U ⬟ 360° 78°, or 282° ii) tan 78° ⬟ 4.704 tan 282° ⬟ 4.704 sin 78° ⬟ 0.978 sin 282° ⬟ 0.978 c) tan u = 0.45 i) tan1(0.45) ⬟ 24° tan U is positive in Quadrant 1, so U ⬟ 24° or in Quadrant 3, so U ⬟ 180° 24°, or 204° ii) cos 24° ⬟ 0.914 cos 204° ⬟ 0.914 sin 24° ⬟ 0.407 sin 204° ⬟ 0.407 18 b) sin u = 0.9 i) sin1(0.9) ⬟ 64° sin U is positive in Quadrant 1, so U ⬟ 64° or in Quadrant 2, so U ⬟ 180° 64°, or 116° ii) tan 64° ⬟ 2.050 tan 116° ⬟ 2.050 cos 64° ⬟ 0.438 cos 116° ⬟ 0.438 d) cos u = ⫺0.3 i) cos1(0.3) ⬟ 73° cos U is negative in Quadrant 2, so U ⬟ 180° 73°, or 107° or in Quadrant 3, so U ⬟ 180° 73°, or 253° ii) sin 107° ⬟ 0.956 sin 253° ⬟ 0.956 tan 107° ⬟ 3.271 tan 253° ⬟ 3.271 6.2 Angles in Standard Position in All Quadrants—Solutions DO NOT COPY. ©P 06_ch06_pre-calculas11_wncp_solution.qxd 5/28/11 2:12 PM Page 19 Home e) sin u = ⫺0.3 i) sin1(0.3) ⬟ 17° sin U is negative in Quadrant 3, so U ⬟ 180° 17°, or 197° or in Quadrant 4, so U ⬟ 360° 17°, or 343° ii) cos 197° ⬟ 0.956 cos 343° ⬟ 0.956 tan 197° ⬟ 0.306 tan 343° ⬟ 0.306 Quit f) tan u = ⫺1.8 i) tan1(1.8) ⬟ 61° tan U is negative in Quadrant 2, so U ⬟ 180° 61°, or 119° or in Quadrant 4, so U ⬟ 360° 61°, or 299° ii) sin 119° ⬟ 0.875 sin 299° ⬟ 0.875 cos 119° ⬟ 0.485 cos 299° ⬟ 0.485 C 20. a) For which values of u is tan u undefined, where 0° ≤ u ≤ 360°? From the table on page 444, tan 90° and tan 270° are undefined. b) Are there any values of u, where 0° ≤ u ≤ 360°, for which sin u or cos u are undefined? Justify your answers. sin U and cos U are defined for all values of U, where 0° ◊ U ◊ 360°, because they are ratios with r in the denominator, and r 0. 21. To the nearest degree, which values of u satisfy each equation for 0° ≤ u ≤ 360°? a) tan u = ⫺tan 60° tan U is negative in Quadrants 2 and 4, so U 180° 60°, or 120°; and U 360° 60°, or 300° ©P DO NOT COPY. b) cos u = sin u In Quadrant 1, where both cos U and sin U are positive, U 45° In Quadrant 3, where both cos U and sin U are negative, U 225° 6.2 Angles in Standard Position in All Quadrants—Solutions 19